Iterative Methods For Solving Linear Matrix Equation And The Convergence Properties

The linear matrix equations arise in many applications such as image restoration, large eigenvalue problems,boundary value problems and model reduction techniques in large-scale dynamical systems.In this thesis,the iterative methods for solving linear matrix equation and the convergence properties are studied.It contains three parts:In Chapter 1,an efficient iterative method is presented to solve the linear matrix equation(?)(X)= E with real matrix X.By this iterative method,the solvability of the linear matrix equation can be determined automatically.When the matrix equation is consistent,then,for any initial matrix X₀,a solution can be obtained within finite iteration steps in the absence of roundoff errors,and the least norm solution can be obtained by choosing a special kind of initial matrix.We also propose an iterative algorithm to obtain the solution or the least norm solution of the consistent matrix system.The given numerical examples demonstrate the efficiency of these two algorithms.In Chapter 2,we study convergence properties of the global orthogonal residual methods for symmetric definite linear matrix equation AX = B.Using the Schur complement and a new matrix product,we present expressions of the norm of the error and the residual.We also derive some useful relations between them.In Chapter 3,we present the global planar conjugate gradient method to solve symmetric indefinite linear matrix equation AX = B.This method can conquer the hard breakdown that may happen when applying the standard global conjugate gradient to such equation.We prove the global convergence for this new method.Finally, numerical examples on test matrices from Harwell-Boeing collection are reported to show the efficiency of this new method.